Find the locus of a complex number `z= x+yi`, satisfying the relation `|3z- 4i| le |3z + 2|`. Illustrate the locus in the Argand plane

Answer

To find the locus of the complex number \( z = x + yi \) satisfying the relation \( |3z - 4i| \leq |3z + 2| \), we will follow these steps:

### Step 1: Substitute \( z \) in the given inequality

We start by substituting \( z = x + yi \) into the inequality:

\[ |3z - 4i| \leq |3z + 2| \]

This becomes:

\[ |3(x + yi) - 4i| \leq |3(x + yi) + 2| \]

### Step 2: Simplify both sides

Now, simplify both sides of the inequality:

Left-hand side:
\[ |3x + 3yi - 4i| = |3x + (3y - 4)i| = \sqrt{(3x)^2 + (3y - 4)^2} \]

Right-hand side:
\[ |3(x + yi) + 2| = |3x + 3yi + 2| = |(3x + 2) + 3yi| = \sqrt{(3x + 2)^2 + (3y)^2} \]

### Step 3: Set up the inequality

Now, we have the inequality:

\[ \sqrt{(3x)^2 + (3y - 4)^2} \leq \sqrt{(3x + 2)^2 + (3y)^2} \]

### Step 4: Square both sides

To eliminate the square roots, we square both sides:

\[ (3x)^2 + (3y - 4)^2 \leq (3x + 2)^2 + (3y)^2 \]

### Step 5: Expand both sides

Expanding both sides gives:

Left-hand side:
\[ 9x^2 + (3y - 4)^2 = 9x^2 + (9y^2 - 24y + 16) \]

Right-hand side:
\[ (3x + 2)^2 + (3y)^2 = (9x^2 + 12x + 4) + 9y^2 \]

### Step 6: Combine and simplify

Now, we combine and simplify the inequality:

\[ 9x^2 + 9y^2 - 24y + 16 \leq 9x^2 + 12x + 4 + 9y^2 \]

Subtract \( 9x^2 + 9y^2 \) from both sides:

\[ -24y + 16 \leq 12x + 4 \]

### Step 7: Rearranging the inequality

Rearranging gives:

\[ 12x + 24y \geq 12 \]

Dividing through by 12:

\[ x + 2y \geq 1 \]

### Step 8: Identify the locus

The locus of the complex number \( z \) is given by the inequality \( x + 2y \geq 1 \). This represents a half-plane in the Argand plane.

### Step 9: Illustrate the locus

To illustrate this in the Argand plane, we can find points on the line \( x + 2y = 1 \):

1. For \( x = 0 \):
\[ 0 + 2y = 1 \implies y = \frac{1}{2} \quad \text{(Point: (0, 0.5))} \]

2. For \( y = 0 \):
\[ x + 2(0) = 1 \implies x = 1 \quad \text{(Point: (1, 0))} \]

### Step 10: Draw the line and shade the region

Plot the points \( (0, 0.5) \) and \( (1, 0) \) on the Argand plane. Draw the line \( x + 2y = 1 \) and shade the region above this line to represent \( x + 2y \geq 1 \).